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Q. If \( 2^x = 3^y = 6^z \), find \( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \)
(a) \( \frac{2}{x} \) (b) \( \frac{2}{y} \) (c) \( \frac{2}{z} \) (d) 1
✏️ Solution
\( 2^x = 3^y = 6^z = k \)
\( x = \frac{\log k}{\log 2},\quad y = \frac{\log k}{\log 3},\quad z = \frac{\log k}{\log 6} \)
\( \frac{1}{x} = \frac{\log 2}{\log k},\quad \frac{1}{y} = \frac{\log 3}{\log k},\quad \frac{1}{z} = \frac{\log 6}{\log k} \)
\( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{\log 2 + \log 3 + \log 6}{\log k} \)
\( = \frac{\log 36}{\log k} \)
\( = \frac{2 \log 6}{\log k} \)
\( = \frac{2}{z} \)
\( \boxed{\frac{2}{z}} \)